
where \(g_X\)  is the gauge coupling in the hidden sector, \(J_X\)  is the hidden sector current and \(\sigma \)  is a pseudoscalar field which is absorbed in a gauge-invariant way via the Stueckelberg mechanism to give mass to the extra neutral gauge boson which we call \(\gamma ^{\prime }\)  (the dark photon). Further one may introduce matter in the hidden sector which is
neutral under \(U(1)_Y\)  but charged under \(U(1)_X\)  [1]}, [2]}.
More generally one may have both kinetic mixing and mass mixing [3]}.
